IndisputableMonolith.ILG.GrowthODE
The ILG.GrowthODE module defines the prefactor for the first-order ILG growth correction in an EdS background. Cosmologists modeling modified gravity effects on linear growth would cite these objects. The definitions arise by substituting the ansatz D = a(1 + B a^α) into the standard growth ODE and simplifying with the ILG kernel.
claimThe prefactor for the growth correction satisfies the first-order ODE obtained by substituting the ansatz $D(a) = a(1 + B a^\alpha)$ into the EdS growth equation, using the ILG kernel $w(k,a) = 1 + C (a/(k \tau_0))^\alpha$ and the variable $X = k \tau_0 / a$.
background
The upstream Kernel module introduces the ILG kernel $w(k,a) = 1 + C \cdot (a/(k \tau_0))^\alpha$. The Reciprocity module defines the dimensionless variable $X = k \tau_0 / a$. These enter the growth ODE that governs the linear density contrast in the presence of the ILG modification to gravity.
The module specializes the general ILG setup to the Einstein-de Sitter background, where the scale factor follows the standard matter-dominated evolution. The DOC_COMMENT states that the prefactor is derived by plugging the ansatz $D = a(1 + B a^\alpha)$ into that ODE.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the growth prefactor and related ODE objects that feed the ISWSign module. The downstream doc-comment notes that a positive driver B combined with negative gravitational potential Phi predicts a negative CMB-LSS cross-correlation at low multipoles, and this growth correction supplies the required f > 1 and ∂lnw > 0 behavior at late time.
scope and limits
- Does not derive the ILG kernel w(k,a).
- Does not treat non-EdS backgrounds.
- Does not compute numerical values of C or alpha.
- Does not address nonlinear corrections.